Colt 1.2.0

## cern.colt.matrix.doublealgo Class Statistic

java.lang.Object
cern.colt.matrix.doublealgo.Statistic

public class Statistic
extends Object

Basic statistics operations on matrices. Computation of covariance, correlation, distance matrix. Random sampling views. Conversion to histograms with and without OLAP cube operators. Conversion to bins with retrieval of statistical bin measures. Also see cern.jet.stat and hep.aida.bin, in particular DynamicBin1D.

Examples:

 A covariance(A) correlation(covariance(A)) distance(A,EUCLID) 4 x 3 matrix 1  2   3 2  4   6 3  6   9 4 -8 -10 3 x 3 matrix  1.25 -3.5 -4.5 -3.5  29   39   -4.5  39   52.5 3 x 3 matrix  1        -0.581318 -0.555492 -0.581318  1         0.999507 -0.555492  0.999507  1 3 x 3 matrix  0        12.569805 15.874508 12.569805  0         4.242641 15.874508  4.242641  0

Version:
1.0, 09/24/99

 Nested Class Summary static interface Statistic.VectorVectorFunction           Interface that represents a function object: a function that takes two argument vectors and returns a single value.

 Field Summary static Statistic.VectorVectorFunction BRAY_CURTIS           Bray-Curtis distance function; Sum( abs(x[i]-y[i]) ) / Sum( x[i]+y[i] ). static Statistic.VectorVectorFunction CANBERRA           Canberra distance function; Sum( abs(x[i]-y[i]) / abs(x[i]+y[i]) ). static Statistic.VectorVectorFunction EUCLID           Euclidean distance function; Sqrt(Sum( (x[i]-y[i])^2 )). static Statistic.VectorVectorFunction MANHATTAN           Manhattan distance function; Sum( abs(x[i]-y[i]) ). static Statistic.VectorVectorFunction MAXIMUM           Maximum distance function; Max( abs(x[i]-y[i]) ).

 Method Summary static DoubleMatrix2D aggregate(DoubleMatrix2D matrix, BinFunction1D[] aggr, DoubleMatrix2D result)           Applies the given aggregation functions to each column and stores the results in a the result matrix. static DynamicBin1D bin(DoubleMatrix1D vector)           Fills all cell values of the given vector into a bin from which statistics measures can be retrieved efficiently. static DoubleMatrix2D correlation(DoubleMatrix2D covariance)           Modifies the given covariance matrix to be a correlation matrix (in-place). static DoubleMatrix2D covariance(DoubleMatrix2D matrix)           Constructs and returns the covariance matrix of the given matrix. static IHistogram2D cube(DoubleMatrix1D x, DoubleMatrix1D y, DoubleMatrix1D weights)           2-d OLAP cube operator; Fills all cells of the given vectors into the given histogram. static IHistogram3D cube(DoubleMatrix1D x, DoubleMatrix1D y, DoubleMatrix1D z, DoubleMatrix1D weights)           3-d OLAP cube operator; Fills all cells of the given vectors into the given histogram. static void demo1()           Demonstrates usage of this class. static void demo2(int rows, int columns, boolean print)           Demonstrates usage of this class. static void demo3(Statistic.VectorVectorFunction norm)           Demonstrates usage of this class. static DoubleMatrix2D distance(DoubleMatrix2D matrix, Statistic.VectorVectorFunction distanceFunction)           Constructs and returns the distance matrix of the given matrix. static IHistogram1D histogram(IHistogram1D histo, DoubleMatrix1D vector)           Fills all cells of the given vector into the given histogram. static IHistogram2D histogram(IHistogram2D histo, DoubleMatrix1D x, DoubleMatrix1D y)           Fills all cells of the given vectors into the given histogram. static IHistogram2D histogram(IHistogram2D histo, DoubleMatrix1D x, DoubleMatrix1D y, DoubleMatrix1D weights)           Fills all cells of the given vectors into the given histogram. static IHistogram3D histogram(IHistogram3D histo, DoubleMatrix1D x, DoubleMatrix1D y, DoubleMatrix1D z, DoubleMatrix1D weights)           Fills all cells of the given vectors into the given histogram. static void main(String[] args)           Benchmarks covariance computation. static DoubleMatrix1D viewSample(DoubleMatrix1D matrix, double fraction, RandomEngine randomGenerator)           Constructs and returns a sampling view with a size of round(matrix.size() * fraction). static DoubleMatrix2D viewSample(DoubleMatrix2D matrix, double rowFraction, double columnFraction, RandomEngine randomGenerator)           Constructs and returns a sampling view with round(matrix.rows() * rowFraction) rows and round(matrix.columns() * columnFraction) columns. static DoubleMatrix3D viewSample(DoubleMatrix3D matrix, double sliceFraction, double rowFraction, double columnFraction, RandomEngine randomGenerator)           Constructs and returns a sampling view with round(matrix.slices() * sliceFraction) slices and round(matrix.rows() * rowFraction) rows and round(matrix.columns() * columnFraction) columns.

 Methods inherited from class java.lang.Object equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

 Field Detail

### EUCLID

public static final Statistic.VectorVectorFunction EUCLID
Euclidean distance function; Sqrt(Sum( (x[i]-y[i])^2 )).

### BRAY_CURTIS

public static final Statistic.VectorVectorFunction BRAY_CURTIS
Bray-Curtis distance function; Sum( abs(x[i]-y[i]) ) / Sum( x[i]+y[i] ).

### CANBERRA

public static final Statistic.VectorVectorFunction CANBERRA
Canberra distance function; Sum( abs(x[i]-y[i]) / abs(x[i]+y[i]) ).

### MAXIMUM

public static final Statistic.VectorVectorFunction MAXIMUM
Maximum distance function; Max( abs(x[i]-y[i]) ).

### MANHATTAN

public static final Statistic.VectorVectorFunction MANHATTAN
Manhattan distance function; Sum( abs(x[i]-y[i]) ).

 Method Detail

### aggregate

public static DoubleMatrix2D aggregate(DoubleMatrix2D matrix,
BinFunction1D[] aggr,
DoubleMatrix2D result)
Applies the given aggregation functions to each column and stores the results in a the result matrix. If matrix has shape m x n, then result must have shape aggr.length x n. Tip: To do aggregations on rows use dice views (transpositions), as in aggregate(matrix.viewDice(),aggr,result.viewDice()).

Parameters:
matrix - any matrix; a column holds the values of a given variable.
aggr - the aggregation functions to be applied to each column.
result - the matrix to hold the aggregation results.
Returns:
result (for convenience only).
Formatter, BinFunction1D, BinFunctions1D

### bin

public static DynamicBin1D bin(DoubleMatrix1D vector)
Fills all cell values of the given vector into a bin from which statistics measures can be retrieved efficiently. Cells values are copied.
Tip: Use System.out.println(bin(vector)) to print most measures computed by the bin. Example:
 Size: 20000 Sum: 299858.02350278624 SumOfSquares: 5399184.154095971 Min: 0.8639113139711261 Max: 59.75331890541892 Mean: 14.992901175139313 RMS: 16.43043540825375 Variance: 45.17438077634358 Standard deviation: 6.721188940681818 Standard error: 0.04752598277592142 Geometric mean: 13.516615397064466 Product: Infinity Harmonic mean: 11.995174297952191 Sum of inversions: 1667.337172700724 Skew: 0.8922838940067878 Kurtosis: 1.1915828121825598 Sum of powers(3): 1.1345828465808412E8 Sum of powers(4): 2.7251055344494686E9 Sum of powers(5): 7.367125643433887E10 Sum of powers(6): 2.215370909100143E12 Moment(0,0): 1.0 Moment(1,0): 14.992901175139313 Moment(2,0): 269.95920770479853 Moment(3,0): 5672.914232904206 Moment(4,0): 136255.27672247344 Moment(5,0): 3683562.8217169433 Moment(6,0): 1.1076854545500715E8 Moment(0,mean()): 1.0 Moment(1,mean()): -2.0806734113421045E-14 Moment(2,mean()): 45.172122057305664 Moment(3,mean()): 270.92018671421 Moment(4,mean()): 8553.8664869067 Moment(5,mean()): 153357.41712233616 Moment(6,mean()): 4273757.570142922 25%, 50% and 75% Quantiles: 10.030074811938091, 13.977982089912224, 18.86124362967137 quantileInverse(mean): 0.559163335012079 Distinct elements & frequencies not printed (too many).

Parameters:
vector - the vector to analyze.
Returns:
a bin holding the statistics measures of the vector.

### correlation

public static DoubleMatrix2D correlation(DoubleMatrix2D covariance)
Modifies the given covariance matrix to be a correlation matrix (in-place). The correlation matrix is a square, symmetric matrix consisting of nothing but correlation coefficients. The rows and the columns represent the variables, the cells represent correlation coefficients. The diagonal cells (i.e. the correlation between a variable and itself) will equal 1, for the simple reason that the correlation coefficient of a variable with itself equals 1. The correlation of two column vectors x and y is given by corr(x,y) = cov(x,y) / (stdDev(x)*stdDev(y)) (Pearson's correlation coefficient). A correlation coefficient varies between -1 (for a perfect negative relationship) to +1 (for a perfect positive relationship). See the math definition and another def. Compares two column vectors at a time. Use dice views to compare two row vectors at a time.

Parameters:
covariance - a covariance matrix, as, for example, returned by method covariance(DoubleMatrix2D).
Returns:
the modified covariance, now correlation matrix (for convenience only).

### covariance

public static DoubleMatrix2D covariance(DoubleMatrix2D matrix)
Constructs and returns the covariance matrix of the given matrix. The covariance matrix is a square, symmetric matrix consisting of nothing but covariance coefficients. The rows and the columns represent the variables, the cells represent covariance coefficients. The diagonal cells (i.e. the covariance between a variable and itself) will equal the variances. The covariance of two column vectors x and y is given by cov(x,y) = (1/n) * Sum((x[i]-mean(x)) * (y[i]-mean(y))). See the math definition. Compares two column vectors at a time. Use dice views to compare two row vectors at a time.

Parameters:
matrix - any matrix; a column holds the values of a given variable.
Returns:
the covariance matrix (n x n, n=matrix.columns).

### cube

public static IHistogram2D cube(DoubleMatrix1D x,
DoubleMatrix1D y,
DoubleMatrix1D weights)
2-d OLAP cube operator; Fills all cells of the given vectors into the given histogram. If you use hep.aida.ref.Converter.toString(histo) on the result, the OLAP cube of x-"column" vs. y-"column" , summing the weights "column" will be printed. For example, aggregate sales by product by region.

Computes the distinct values of x and y, yielding histogram axes that capture one distinct value per bin. Then fills the histogram.

Example output:

 Cube:    Entries=5000, ExtraEntries=0    MeanX=4.9838, RmsX=NaN    MeanY=2.5304, RmsY=NaN    xAxis: Min=0, Max=10, Bins=11    yAxis: Min=0, Max=5, Bins=6 Heights:       | X       | 0 1 2 3 4 5 6 7 8 9 10 | Sum ---------------------------------------------------------- Y 5 | 30 53 51 52 57 39 65 61 55 49 22 | 534   4 | 43 106 112 96 92 94 107 98 98 110 47 | 1003   3 | 39 134 87 93 102 103 110 90 114 98 51 | 1021   2 | 44 81 113 96 101 86 109 83 111 93 42 | 959   1 | 54 94 103 99 115 92 98 97 103 90 44 | 989   0 | 24 54 52 44 42 56 46 47 56 53 20 | 494 ----------------------------------------------------------   Sum | 234 522 518 480 509 470 535 476 537 493 226 |

Returns:
the histogram containing the cube.
Throws:
IllegalArgumentException - if x.size() != y.size() || y.size() != weights.size().

### cube

public static IHistogram3D cube(DoubleMatrix1D x,
DoubleMatrix1D y,
DoubleMatrix1D z,
DoubleMatrix1D weights)
3-d OLAP cube operator; Fills all cells of the given vectors into the given histogram. If you use hep.aida.ref.Converter.toString(histo) on the result, the OLAP cube of x-"column" vs. y-"column" vs. z-"column", summing the weights "column" will be printed. For example, aggregate sales by product by region by time.

Computes the distinct values of x and y and z, yielding histogram axes that capture one distinct value per bin. Then fills the histogram.

Returns:
the histogram containing the cube.
Throws:
IllegalArgumentException - if x.size() != y.size() || x.size() != z.size() || x.size() != weights.size().

### demo1

public static void demo1()
Demonstrates usage of this class.

### demo2

public static void demo2(int rows,
int columns,
boolean print)
Demonstrates usage of this class.

### demo3

public static void demo3(Statistic.VectorVectorFunction norm)
Demonstrates usage of this class.

### distance

public static DoubleMatrix2D distance(DoubleMatrix2D matrix,
Statistic.VectorVectorFunction distanceFunction)
Constructs and returns the distance matrix of the given matrix. The distance matrix is a square, symmetric matrix consisting of nothing but distance coefficients. The rows and the columns represent the variables, the cells represent distance coefficients. The diagonal cells (i.e. the distance between a variable and itself) will be zero. Compares two column vectors at a time. Use dice views to compare two row vectors at a time.

Parameters:
matrix - any matrix; a column holds the values of a given variable (vector).
distanceFunction - (EUCLID, CANBERRA, ..., or any user defined distance function operating on two vectors).
Returns:
the distance matrix (n x n, n=matrix.columns).

### histogram

public static IHistogram1D histogram(IHistogram1D histo,
DoubleMatrix1D vector)
Fills all cells of the given vector into the given histogram.

Returns:
histo (for convenience only).

### histogram

public static IHistogram2D histogram(IHistogram2D histo,
DoubleMatrix1D x,
DoubleMatrix1D y)
Fills all cells of the given vectors into the given histogram.

Returns:
histo (for convenience only).
Throws:
IllegalArgumentException - if x.size() != y.size().

### histogram

public static IHistogram2D histogram(IHistogram2D histo,
DoubleMatrix1D x,
DoubleMatrix1D y,
DoubleMatrix1D weights)
Fills all cells of the given vectors into the given histogram.

Returns:
histo (for convenience only).
Throws:
IllegalArgumentException - if x.size() != y.size() || y.size() != weights.size().

### histogram

public static IHistogram3D histogram(IHistogram3D histo,
DoubleMatrix1D x,
DoubleMatrix1D y,
DoubleMatrix1D z,
DoubleMatrix1D weights)
Fills all cells of the given vectors into the given histogram.

Returns:
histo (for convenience only).
Throws:
IllegalArgumentException - if x.size() != y.size() || x.size() != z.size() || x.size() != weights.size().

### main

public static void main(String[] args)
Benchmarks covariance computation.

### viewSample

public static DoubleMatrix1D viewSample(DoubleMatrix1D matrix,
double fraction,
RandomEngine randomGenerator)
Constructs and returns a sampling view with a size of round(matrix.size() * fraction). Samples "without replacement" from the uniform distribution.

Parameters:
matrix - any matrix.
randomGenerator - a uniform random number generator; set this parameter to null to use a default generator seeded with the current time.
Returns:
the sampling view.
Throws:
IllegalArgumentException - if ! (0 <= rowFraction <= 1 && 0 <= columnFraction <= 1).
RandomSampler

### viewSample

public static DoubleMatrix2D viewSample(DoubleMatrix2D matrix,
double rowFraction,
double columnFraction,
RandomEngine randomGenerator)
Constructs and returns a sampling view with round(matrix.rows() * rowFraction) rows and round(matrix.columns() * columnFraction) columns. Samples "without replacement". Rows and columns are randomly chosen from the uniform distribution. Examples:
 matrix rowFraction=0.2 columnFraction=0.2 rowFraction=0.2 columnFraction=1.0 rowFraction=1.0 columnFraction=0.2 10 x 10 matrix  1  2  3  4  5  6  7  8  9  10 11 12 13 14 15 16 17 18 19  20 21 22 23 24 25 26 27 28 29  30 31 32 33 34 35 36 37 38 39  40 41 42 43 44 45 46 47 48 49  50 51 52 53 54 55 56 57 58 59  60 61 62 63 64 65 66 67 68 69  70 71 72 73 74 75 76 77 78 79  80 81 82 83 84 85 86 87 88 89  90 91 92 93 94 95 96 97 98 99 100 2 x 2 matrix 43 50 53 60 2 x 10 matrix 41 42 43 44 45 46 47 48 49  50 91 92 93 94 95 96 97 98 99 100 10 x 2 matrix  4  8 14 18 24 28 34 38 44 48 54 58 64 68 74 78 84 88 94 98

Parameters:
matrix - any matrix.
rowFraction - the percentage of rows to be included in the view.
columnFraction - the percentage of columns to be included in the view.
randomGenerator - a uniform random number generator; set this parameter to null to use a default generator seeded with the current time.
Returns:
the sampling view.
Throws:
IllegalArgumentException - if ! (0 <= rowFraction <= 1 && 0 <= columnFraction <= 1).
RandomSampler

### viewSample

public static DoubleMatrix3D viewSample(DoubleMatrix3D matrix,
double sliceFraction,
double rowFraction,
double columnFraction,
RandomEngine randomGenerator)
Constructs and returns a sampling view with round(matrix.slices() * sliceFraction) slices and round(matrix.rows() * rowFraction) rows and round(matrix.columns() * columnFraction) columns. Samples "without replacement". Slices, rows and columns are randomly chosen from the uniform distribution.

Parameters:
matrix - any matrix.
sliceFraction - the percentage of slices to be included in the view.
rowFraction - the percentage of rows to be included in the view.
columnFraction - the percentage of columns to be included in the view.
randomGenerator - a uniform random number generator; set this parameter to null to use a default generator seeded with the current time.
Returns:
the sampling view.
Throws:
IllegalArgumentException - if ! (0 <= sliceFraction <= 1 && 0 <= rowFraction <= 1 && 0 <= columnFraction <= 1).